vline

• Let \varepsilon > 0.

  \varepsilon is fixed We have \begin{aligned} X_t = \sqrt{\alpha_t}X_{t-1} + \sqrt{1-\alpha_t} \varepsilon_{t}. \end{aligned} To have the concepts of independence and noise, we need to have probability measures. In the following text, we use lowercase q(x) to denote the density of the probability measure \mathbf{Q} corresponding to the random variable X. Others (e.g., q(x_t),p_{\theta}(x_t)) are the same. (p_{\theta} corresponds to \mathbf{P}_{\theta}). We also use q(x_{0:t}) to denote the density of (X_0,X_1,\cdots,X_t):=X_{0:t} for the probability measure \mathbf{Q}. Others are the same.

  Suppose q_{\text{want}}(x_0) is the density of X_0 we want to pursue. We do not know what q_{\text{want}}(x_0) is. We only have some eligible images (discrete data) with mass function {\color{blue}{q(x_0)}}. When this discrete data large, q(x_0)\approx q_{\text{want}}(x_0) in some sense of distribution. Our goal is to find a density p(x_0) of X_0 such that p(x_0)\approx q_{\text{want}}(x_0) in some sense of distribution.

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ASD

CCC

Once Loop Reflect

Code

III

Getting up

• Turn off alarm
• Get out of bed

asdadas

Breakfast

• Eat eggs
• Drink coffee

Dinner

• Eat spaghetti
• Drink wine

Going to sleep

• Get in bed
• Count sheep

Going to sleep

• Get in bed
• Count sheep